Finite Group Databasebeta

← #14418621

Subgroups — #14418621

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65851C211yesmaximal, characteristic
2#11751100C211yesmaximal, characteristic
3C5×#4997C211yesmaximal, characteristic
4#11748433C211yesmaximal, characteristic
5#11748431C211yesmaximal, characteristic
6C2×#90952C211yesmaximal, characteristic
7#11748318C211yesmaximal, characteristic
8#11746861C214yesmaximal, complemented
9#11746847C214yesmaximal, complemented
10#10610274—31nomaximal, Hall
11C3×C5×#117C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C22×C3⋊#168C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C22×C5×C3⋊C8C2211yescharacteristic
16C3×#4072C2211yescharacteristic
17C2×C3×C4×Dic5C2211yescharacteristic
18C3×#4036C412yes—
19C3×#4034C412yes—
20C3×#4024C414yes—
21C3×C4×Dic10C2214yes—
22C4⋊(C3⋊#168)C2214yes—
23C5×#803C2212yes—
24C4×C5×C3⋊C8C2212yes—
25#90982C2214yes—
26#90982C2214yes—
27#90952C2212yes—
28#90952C2212yes—
29#90952C2212yes—
30Dic5×C3⋊C8C2212yes—
31#83329—51nomaximal, Hall
32#65851D311yescharacteristic
33C5×#405—31no—
34#65408—31no—
35#65407—31no—
36#65390—31no—
37C2×#3947—31no—
38#65288—31no—
39#64990—34no—
40#64968—34no—
41C22×C3×C4×C5C2311yescharacteristic
42C22×C3×Dic5C2×C412yes—
43C2×C3×Dic10C2×C414yes—
44C3×C5×C4⋊C4C2×C412yes—
45C3×C5×C22⋊C4C2×C412yes—
46C3×C42×C5C2312yes—
47#6710—22no—
48C2×C3⋊#168—22no—
49C2×C3⋊#168C2312yes—
50C2×C3⋊#168C2312yes—
51C5×#208—22no—
52C2×C5×C3⋊C8Q814yescomplemented
53C2×C5×C3⋊C8C2312yes—
54C2×C5×C3⋊C8C2311yescharacteristic
55C2×C5×C3⋊C8C2311yescharacteristic
56C3×C22⋊Dic5C2×C412yes—
57C3×C4⋊Dic5C2×C414yes—
58C3×C4⋊Dic5C2×C412yes—
59C3×#683C2×C414yes—
60C3×#683C2×C414yes—
61C3×C4×Dic5C2×C411yescharacteristic
62C3×C4×Dic5C2×C411yescharacteristic
63C3×C4×Dic5C2312yes—
64C3×C4×Dic5C2312yes—
65C3×#533—51no—
66#5024—54no—
67#5024—54no—
68#4999—51no—
69#4999—51no—
70#4997—51no—
71#4997D511yescharacteristic
72#4924—51no—
73C2×#803—51no—
74C2×C5×M16—31no—
75C22×C5×C8—31no—
76C5×#117C2×D311yescharacteristic
77#4072C2×D311yescharacteristic
78C2×C4×Dic5C2×D311yescharacteristic
79C2×#681—31no—
80C22×#168—31no—
81#4036Dic312yes—
82#4034Dic312yes—
83#4024Dic314yes—
84C4×Dic10C2×D314yes—
85C5×#97—32no—
86C4×C5×C8—32no—
87#3947—32no—
88#3947—32no—
89#3947—32no—
90#3946—34no—
91#3946—34no—
92C8×Dic5—32no—
93C4⋊#168—34no—
94#2961—151noSylow
95C23×C3×C5C22×C411yescharacteristic
96C2×C3×C4×C5C22×C414yes—
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×C3×C4×C5C2×Q812yes—
99C2×C3×C4×C5#4112yes—
100C2×C3×C4×C5C2411yescharacteristic
101C2×C3×Dic5—42no—
102C2×C3×Dic5C22×C412yes—
103C2×C3×Dic5C22×C412yes—
104C2×C3×Dic5C22×C414yes—
105C3×Dic10—44no—
106C3⋊#168—22no—
107C3⋊#168—22no—
108C5×C3⋊C8—22no—
109C5×C3⋊C8C2×Q812yes—
110C5×C3⋊C8C2×Q812yes—
111C3×C4⋊Q8—52no—
112C3×#125—52no—
113C3×C22⋊Q8—54no—
114C3×C4×Q8—54no—
115C3×#117—51no—
116C3×#117C2×D511yescharacteristic
117C2×C3×C42—51no—
118C2×#208—51no—
119C2×#208C2×D511yescharacteristic
120C22×C3⋊C8—51no—
121C22×C3⋊C8C2×D511yescharacteristic
122C4⋊(C3⋊C8)—54no—
123C4⋊(C3⋊C8)—54no—
124C4⋊(C3⋊C8)—54no—
125#803—52no—
126#803—52no—
127#803—52no—
128#803C2×D512yes—
129C4×C3⋊C8—52no—
130C4×C3⋊C8C2×D512yes—
131C22×C4×C5C22×D311yescharacteristic
132C22×Dic5C2×Dic312yes—
133C2×Dic10C2×Dic314yes—
134C5×M16—62no—
135C2×C5×C8—34no—
136C2×C5×C8—31no—
137C2×C5×C8—31no—
138C2×C5×C8—32no—
139C5×C4⋊C4C2×Dic312yes—
140C5×C22⋊C4C2×Dic312yes—
141C42×C5C22×D312yes—
142C22⋊Dic5C2×Dic312yes—
143C4⋊Dic5C2×Dic314yes—
144C4⋊Dic5C2×Dic312yes—
145#683C2×Dic314yes—
146#683C2×Dic314yes—
147C4×Dic5C2×Dic311yescharacteristic
148C4×Dic5C2×Dic311yescharacteristic
149C4×Dic5C22×D312yes—
150C4×Dic5C22×D312yes—
151#681—62no—
152C2×#168—32no—
153C2×#168—32no—
154C2×#168—62no—
155#533—51no—
156#446—154no—
157#446—154no—
158#424—151no—
159#424—151no—
160#423—151no—
161#405—31no—
162#405—151no—
163C2×#97—151no—
164C22×C3×C5—22no—
165C22×C3×C5C4×Q812yes—
166C22×C3×C5C23×C411yescharacteristic
167C3×C4×C5—24no—
168C3×C4×C5C4×Q812yes—
169C3×C4×C5C22×Q811yescharacteristic
170C3×C4×C5C2×#4111yescharacteristic
171C3×Dic5—22no—
172C3×Dic5—24no—
173C3×Dic5—22no—
174C2×C3×Q8—54no—
175C22×C3×C4—52no—
176C22×C3×C4C22×D511yescharacteristic
177C3×C4⋊C4—54no—
178C3×C4⋊C4—54no—
179C3×C4⋊C4—52no—
180C3×C4⋊C4—54no—
181C3×C4⋊C4C4×D512yes—
182C3×C22⋊C4—52no—
183C3×C22⋊C4C4×D512yes—
184C3×C42—51no—
185C3×C42—51no—
186C3×C42—52no—
187C3×C42—52no—
188C3×C42C22×D512yes—
189#208—22no—
190#208—102no—
191C2×C3⋊C8—52no—
192C2×C3⋊C8—52no—
193C2×C3⋊C8—102no—
194C2×C3⋊C8Dic1014yescomplemented
195C2×C3⋊C8C22×D512yes—
196C2×C3⋊C8C22×D511yescharacteristic
197C2×C3⋊C8C22×D511yescharacteristic
198C23×C5C22×Dic311yescharacteristic
199C2×C4×C5D3×Q812yes—
200C2×C4×C5#23912yes—
201C2×C4×C5C22×Dic314yes—
202C2×C4×C5C22×Dic311yescharacteristic
203C2×C4×C5C23×D311yescharacteristic
204C2×Dic5—42no—
205C2×Dic5C22×Dic312yes—
206C2×Dic5C22×Dic312yes—
207C2×Dic5C22×Dic314yes—
208Dic10—44no—
209C5×C8—32no—
210C5×C8—32no—
211C5×C8—62no—
212#168—62no—
213#168—62no—
214C2×M16—31no—
215C2×M16—151no—
216C22×C8—31no—
217C22×C8—151no—
218C4⋊Q8—52no—
219#125—52no—
220C22⋊Q8—54no—
221C4×Q8—54no—
222#117—51no—
223#117D3×D511yescharacteristic
224C2×C42—51no—
225C4⋊C8—154no—
226C4⋊C8—154no—
227C4⋊C8—154no—
228#97—32no—
229#97—152no—
230#97—152no—
231#97—152no—
232C4×C8—32no—
233C4×C8—152no—
234C2×C3×C5—22no—
235C2×C3×C5C2×C4×Q811yescharacteristic
236C2×C3×C5#56812yes—
237C23×C3C2×C4×D511yescharacteristic
238C3×Q8—204no—
239C2×C3×C4—52no—
240C2×C3×C4—52no—
241C2×C3×C4—54no—
242C2×C3×C4—202no—
243C2×C3×C4C2×Dic1012yes—
244C2×C3×C4C2×C4×D514yes—
245C2×C3×C4C2×C4×D511yescharacteristic
246C2×C3×C4#70912yes—
247C2×C3×C4C23×D511yescharacteristic
248C3⋊C8—22no—
249C3⋊C8—102no—
250C3⋊C8—102no—
251C3⋊C8C2×Dic1012yes—
252C3⋊C8C2×Dic1012yes—
253C22×C5—22no—
254C22×C5Q8×Dic312yes—
255C22×C5C23×Dic311yescharacteristic
256C4×C5—24no—
257C4×C5Q8×Dic312yes—
258C4×C5C2×D3×Q811yescharacteristic
259C4×C5C2×#23911yescharacteristic
260Dic5—22no—
261Dic5—24no—
262Dic5—22no—
263C2×Q8—54no—
264C22×C4—52no—
265C22×C4C2×D3×D511yescharacteristic
266M16—62no—
267M16—302no—
268C2×C8—34no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—32no—
272C2×C8—152no—
273C2×C8—152no—
274C2×C8—302no—
275C4⋊C4—52no—
276C4⋊C4—54no—
277C4⋊C4—54no—
278C4⋊C4—54no—
279C4⋊C4D5×Dic312yes—
280C22⋊C4—52no—
281C22⋊C4D5×Dic312yes—
282C42—51no—
283C42—51no—
284C42—52no—
285C42—52no—
286C42C2×D3×D512yes—
287C3×C5#296111yescharacteristic, complemented, Hall
288C22×C3—22no—
289C22×C3C4×Dic1012yes—
290C22×C3C22×C4×D511yescharacteristic
291C3×C4—24no—
292C3×C4—102no—
293C3×C4—102no—
294C3×C4—104no—
295C3×C4C4×Dic1012yes—
296C3×C4C22×Dic1011yescharacteristic
297C3×C4C2×#70911yescharacteristic
298C2×C5—22no—
299C2×C5C2×Q8×Dic311yescharacteristic
300C2×C5#581912yes—
301C23C2×D5×Dic311yescharacteristic
302Q8—204no—
303C2×C4—52no—
304C2×C4—52no—
305C2×C4—54no—
306C2×C4—202no—
307C2×C4D3×Dic1012yes—
308C2×C4#676912yes—
309C2×C4C2×D5×Dic314yes—
310C2×C4C2×D5×Dic311yescharacteristic
311C2×C4C22×D3×D511yescharacteristic
312C8—32no—
313C8—32no—
314C8—62no—
315C8—302no—
316C8—302no—
317C2×C3—22no—
318C2×C3C2×C4×Dic1011yescharacteristic
319C2×C3#6608412yes—
320C5#8332911yescharacteristic, complemented, Sylow
321C22—22no—
322C22Dic3×Dic1012yes—
323C22C22×D5×Dic311yescharacteristic
324C4—24no—
325C4—102no—
326C4—102no—
327C4—104no—
328C4Dic3×Dic1012yes—
329C4C2×D3×Dic1011yescharacteristic
330C4C2×#676911yescharacteristic
331C3#1061027411yescharacteristic, complemented, Sylow
332C2—22no—
333C2C2×Dic3×Dic1011yescharacteristic
334C2#1175529612yes—
subgroupG/H
center, FrattiniC2×C4C22×D3×D5
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×#117C22
prime radicalC23×C3×C5C22×C4
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2961
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ #117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. #11746861C2semidirect
9. #11746847C2semidirect
52. C2×C5×C3⋊C8Q8semidirect
194. C2×C3⋊C8Dic10semidirect
287. C3×C5#2961semidirect
320. C5#83329semidirect
331. C3#10610274semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#658512
#117511002
C5×#49972
#117484332
#117484312
C2×#909522
#117483182
#117468612
#117468472
#106102743
#833295